Well, assuming Richard Parker is not **Richard Parker’s ** real name, but rather a real person’s fictional Doper name, then both **Richard Parker ** the fictional Doper and Richard Parker the fictional tiger are both fictional.
Or something…
That’s just the thing. The name is not what makes a thing fictional, it is the existence of the referent. If I call my couch “a giant brainsucking monster from outerspace,” then a giant brainsucking monster from outerspace exists. Some things have longer names than others, like, for example, “tiger that stows away in lifeboat.”
(Or maybe I was just making a Life of Pi plot-related joke.)
I believe the version with the tiger and the gorilla and the hyena. 
Richard Parker was real, man- he was real.